cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A104210 Positive integers divisible by at least 2 consecutive primes.

Original entry on oeis.org

6, 12, 15, 18, 24, 30, 35, 36, 42, 45, 48, 54, 60, 66, 70, 72, 75, 77, 78, 84, 90, 96, 102, 105, 108, 114, 120, 126, 132, 135, 138, 140, 143, 144, 150, 154, 156, 162, 165, 168, 174, 175, 180, 186, 192, 195, 198, 204, 210, 216, 221, 222, 225, 228, 231, 234, 240
Offset: 1

Views

Author

Leroy Quet, Mar 13 2005

Keywords

Comments

If a perfect square is in this sequence, then so is its square root (e.g., 144 and 12). - Alonso del Arte, May 07 2012
The numbers of terms not exceeding 10^k, for k=1,2,..., are 1, 22, 242, 2456, 24632, 246414, 2464272, 24643281, 246433426, ... Apparently, the asymptotic density of this sequence is 0.24643... - Amiram Eldar, Apr 10 2021

Examples

			35 is divisible by both 5 and 7, and 5 and 7 are consecutive primes.
77 is divisible by both 7 and 11, and 7 and 11 are consecutive primes.
110 is not in the sequence because, although it is divisible by 2, 5 and 11, it is not divisible by 3 or 7.

Links

Crossrefs

Cf. A003961, A296210 (characteristic function), A319630 (complement), A379230 [= A252748(a(n))].
Positions of terms larger than 1 in A300820 and in A322361.
Subsequences: A006094, A349169 (conjectured, after its initial 1), A349176, A355527 (squarefree terms), A372566, A378884, A379232.

Programs

  • Maple
    N:= 1000: # for terms <= N
R:= {}:
p:= 2:
do
  q:= p; p:= nextprime(p);
  if p*q > N then break fi;
  R:= R union {seq(i,i=p*q..N,p*q)}
od:
sort(convert(R,list)); # Robert Israel, Apr 13 2020
  • Mathematica
    fQ[n_] := Block[{lst = PrimePi /@ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]}, Count[ Drop[lst, 1] - Drop[lst, -1], 1] > 0]; Select[ Range[244], fQ[ # ] &] (* Robert G. Wilson v, Mar 16 2005 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
is_A104210(n) = (gcd(n,A003961(n))>1); \\ Antti Karttunen, Dec 24 2024

Formula

{k such that gcd(k, A003961(k)) > 1}. - Antti Karttunen, Dec 24 2024

Extensions

More terms from Robert G. Wilson v, Mar 16 2005

A378885 Numbers that are divisible by at least three different primes and the smallest three of them are consecutive primes.

Original entry on oeis.org

30, 60, 90, 105, 120, 150, 180, 210, 240, 270, 300, 315, 330, 360, 385, 390, 420, 450, 480, 510, 525, 540, 570, 600, 630, 660, 690, 720, 735, 750, 780, 810, 840, 870, 900, 930, 945, 960, 990, 1001, 1020, 1050, 1080, 1110, 1140, 1155, 1170, 1200, 1230, 1260, 1290
Offset: 1

Views

Author

Amiram Eldar, Dec 09 2024

Keywords

Comments

All the positive multiples of 30 (A249674 \ {0}) are terms.
Numbers k such that A151800(A020639(k)) | k and also A101300(A020639(k)) | k.
The asymptotic density of this sequence is Sum_{k>=1} (Product_{j=1..k-1} (1-1/prime(j))) / (prime(k)*prime(k+1)*prime(k+2)) = 0.03943839735407432193784... .

Examples

			60 = 2^2 * 3 * 5 is a term since 2, 3 and 5 are consecutive primes.
770 = 2 * 5 * 7 * 11 is not a term since its smallest prime divisor is 2 and it is not divisible by 3, the prime next to 2.
1365 = 3 * 5 * 7 * 13 is a term since 3, 5 and 7 are consecutive primes.

Links

Crossrefs

Subsequence of A000977.
Subsequences: A046301, A378884.
Cf. A020639, A101300, A151800, A249674.

Programs

  • Mathematica
    q[k_] := Module[{p = FactorInteger[k][[;; , 1]]}, Length[p] > 2 && p[[2]] == NextPrime[p[[1]]] && p[[3]] == NextPrime[p[[2]]]]; Select[Range[1300], q]
  • PARI
    is(k) = if(k == 1, 0, my(p = factor(k)[,1]); #p > 2 && p[2] == nextprime(p[1]+1) && p[3] == nextprime(p[2]+1));

A378886 The number of consecutive primes in the prime factorization of n starting from the smallest prime dividing n; a(1) = 0.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3
Offset: 1

Views

Author

Amiram Eldar, Dec 10 2024

Keywords

Comments

First differs from A300820 at n = 70 = 2 * 5 * 7: A300820(70) = 2 while a(70) = 1.

Examples

			a(42) = 2 since 42 = 2 * 3 * 7 and 2 and 3 are 2 consecutive primes.
a(28) = 1 since 28 = 2^2 * 7 and 3 is not a divisor of 28.
a(4095) = 3 since 4095 = 3^2 * 5 * 7 * 13 and 3, 5 and 7 are 3 consecutive primes.

Links

Crossrefs

Cf. A001221, A020639, A073491, A276084, A300820, A378884, A378885.

Programs

  • Mathematica
    a[n_] := Module[{p = FactorInteger[n][[;; , 1]], c = 1, q}, q = p[[1]]; Do[q = NextPrime[q]; If[q == p[[i]], c++, Break[]], {i, 2, Length[p]}]; c]; a[1] = 0; Array[a, 100]
  • PARI
    a(n) = if(n == 1, 0, my(p = factor(n)[,1], c = 1, q); q = p[1]; for(i = 2, #p, q = nextprime(q+1); if(q == p[i], c++, break)); c);

Formula

a(n) >= A276084(n).
a(n) <= A300820(n).
a(n) = A001221(n) if and only if n is in A073491.
a(n) >= 1 for n >= 2.
a(n) >= 2 if and only if n is in A378884.
a(n) >= 3 if and only if n is in A378885.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * (d(k) - d(k+1)) = 1.2630925015039..., where d(1) = 1 and d(k) = Sum_{i>=1} (Product_{j=1..i-1} (1-1/prime(j)))/(Product_{j=0..k-1} prime(i+j)), for k >= 2. d(k) is the asymptotic density of numbers m for which a(m) >= k.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.